The polynomial-error conjecture for higher-dimensional partitions

For positive integers $n$ and $d$, let $\Omega^n_d$ be the effective classes defined by

$$\sum_{d\geq 0}[\operatorname{Hilb}^d({\mathbb A}^n)_0]t^d=\operatorname{Exp}\left(\sum_{d>0}\Omega^n_dt^d\right),$$

and let $\chi$ denote the Euler characteristic. Polynomial-error conjecture. For every $d\geq 6$, there exists an irreducible polynomial $r_d(t)\in{\mathbb Q}[t]$ of degree $d-6$ such that

$$\binom{d+n-3}{n-2}-\chi(\Omega^n_d)=\binom{n}{4}r_d(n)$$

for all $n\geq 1$.

The claim describes a regularity in the discrepancy between MacMahon’s proposed product and the actual generating function for higher-dimensional partitions, as measured by Euler characteristics of the motivic classes.